Syllabus

Click below to expand the details for each class, including times and locations, the lecture videos you should watch in advance, the review questions you should solve in advance, and the problem sets you should complete after each class. Please note that all classes will take place at Lady Margaret Hall.

Date, Time, and Location

Thursday, 3:30-6pm, MT Week 3, Location TBC

Videos

  • P0: Introduction
  • P1: Probability Basics
    • Probabilities as set functions; scaled-Venn representation (\(\Omega\) as the unit square)
    • Complements, intersections (“and”), unions (“or”) of events
    • Building blocks: sample space, event, event space \(\mathcal{F}\)
    • Worked sample/event spaces: fair die, coin toss, lightbulb lifetime
    • The three (Kolmogorov) axioms; disjoint events and additivity
    • The four rules: complement, union of two events, Boole/union bound, logical consequence (monotonicity), with the complement-rule derivation
  • P2: Conditional Probability
    • Conditional probability \(P(A|B)\) as information shrinking the sample space
    • Kolmogorov definition \(P(A|B)=P(A\cap B)/P(B)\); de Finetti multiplication rule
    • Venn-diagram intuition (conditioning rescales \(B\) to the unit circle)
    • Revised (conditional) probability axioms
    • Asymmetry: \(P(A|B)\neq P(B|A)\)
    • Law of Total Probability over a partition
  • P3: Independence & Bayes’ Rule
    • Independence via \(P(A\cap B)=P(A)P(B)\) and \(P(A|B)=P(A)\); symmetry
    • Bayes’ Rule: statement, derivation, historical background
    • Law of Total Probability for the denominator
    • Prior / likelihood / evidence / posterior vocabulary
    • Worked example: doping/EPO test and the base-rate problem
    • Sally Clark case: independence fallacy and the prosecutor’s fallacy

Quiz

Before class, complete the review questions for class 1. These will be the basis of the quiz at the start of class.

Flashcards

Download the Class 1 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number one.

Date, Time, and Location

Thursday, 3:30-6pm, MT Week 5, Location TBC

Videos

  • D0: Data & Variables Intro
  • D1: Data Basics
    • Variable types: categorical (binary, ordinal) vs. numerical (discrete, continuous)
    • Support of a variable; univariate vs. multivariate
    • Cross-sectional data \(\{X_i\}_{i=1,\dots,N}\); geometric depiction (number line, 2D/3D scatter)
    • Time-series data \(\{X_t\}_{t=1,\dots,T}\); a time series as one observation on a \(T\)-dimensional variable
    • Panel data; balanced vs. unbalanced
  • D2: Probability Mass Functions
    • PMF definition for discrete variables; frequentist (relative-frequency) interpretation
    • Representations: formula, table, stem (“lollipop”) plot
    • PMF properties: \(f(x)=\mathbb{P}(X=x)\ge 0\), \(\sum_x f(x)=1\); support, \(X\) vs. \(x\)
    • Bernoulli distribution; categorical/“multinoulli”; the Iverson bracket
    • Summation (\(\Sigma\)) and product (\(\Pi\)) notation
    • Binomial distribution: PMF, parameters \(N\) and \(p\), shape
  • D3: Probability Density Functions
    • PDF as a limit; the continuous analogue of the PMF (smoothed histogram)
    • Properties: \(f(x)\ge 0\) and \(\int f(x)\,dx=1\)
    • Subtleties: \(\mathbb{P}(X=x)=0\); \(f(x)\) is not a probability
    • Continuous Uniform: \(f(x)=1/(b-a)\)
    • Normal (Gaussian): density; \(\mu\) controls location, \(\sigma\) spread
    • Probabilities defined over intervals, not points
  • D4: Cumulative Distribution Functions
    • CDF \(F(x)=\mathbb{P}(X\le x)\); right-continuity
    • Discrete CDFs as step functions; continuous CDFs (Uniform, Normal S-curve)
    • Properties: bounds, monotonicity, \(\mathbb{P}(X>x)=1-F(x)\), \(F(b)-F(a)\)
    • PDF–CDF link: \(f=dF/dx\), \(F=\int_{-\infty}^x f\) (FTC)
    • Quantiles from the CDF; reading Standard Normal tables with interpolation
    • Key critical values: \(\pm1.645\) (90%), \(\pm1.96\) (95%), \(\pm2.58\) (99%)
  • D5: Standardization
    • Standard Normal \(N(0,1)\) and the need to standardize general \(N(\mu,\sigma^2)\)
    • Adding a constant (location shift) vs. multiplying (scales location and spread)
    • Standardization \(Z=(X-\mu)/\sigma\sim N(0,1)\), derived in two steps
    • Computing \(\mathbb{P}(X\le x)\) by standardizing; left-tail, right-tail, interval examples
    • Quantiles: look up the \(N(0,1)\) quantile, then de-standardize
  • D6: Multivariate Distributions
    • Bivariate/multivariate data; joint CDF and joint PDF/PMF
    • Marginal distributions from the joint
    • Independence: joint factors into the product of marginals
    • Conditional distributions as scaled slices through the joint
    • Generalization to higher dimensions
Warning

Coming soon — joint distributions worked example. The lecture covers the ideas geometrically. We’ll add a worked example / short handout on the discrete table-based machinery: computing marginals, conditional PMFs, and covariance directly from a joint PMF table.

Quiz

Before class, complete the review questions for class 2. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 1.

Flashcards

Download the Class 2 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number two.

Date, Time, and Location

Thursday, 3:30-6pm, MT Week 7, Location TBC

Videos

  • M0: Intermission
  • M1: Expected Values
    • Expected value (discrete sum vs. continuous integral) as a probability-weighted average
    • Mean need not lie in the support or be the most probable value
    • Means of Uniform \((a+b)/2\) and Normal \(\mu\); expectation of a binary variable equals \(p\)
    • \(\mathbb{E}[g(X)]\) for a function of a variable
    • Rules: affine transformations, additivity (linearity), multiplicativity under independence
    • Jensen’s inequality (concave vs. convex); the heavy-tailed Cauchy (no mean)
  • M2: Variance
    • Variance \(=\mathbb{E}([X-\mathbb{E}X]^2)=\mathbb{E}(X^2)-[\mathbb{E}X]^2\); non-negativity
    • Discrete vs. continuous formulas; standard deviation and units
    • Variance of Uniform \((b-a)^2/12\), Normal \(\sigma^2\), binary \(p(1-p)\)
    • Standardization \(Z=(X-\mu)/\sigma\sim N(0,1)\)
    • The empirical (68–95–99.7) rule
  • M3: Conditional Expectations
    • Conditional expectation \(\mathbb{E}(Y|X)\) as the mean of a conditional distribution
    • Conditional means for prediction; gender wage gap as \(\mathbb{E}[Y|X=1]-\mathbb{E}[Y|X=0]\)
    • Formal definition (discrete and continuous)
    • Properties: pull-out rule, independence, Law of Iterated Expectations
    • \(\mathbb{E}[Y|X=x]\) (a point) vs. \(\mathbb{E}[Y|X]\) (a function)
  • M4: Correlation, Covariance, & Independence
    • Covariance: definition, sign interpretation, scatter-plot intuition
    • Properties: symmetry, \(\text{Cov}(X,X)=\text{Var}(X)\), \(\text{Cov}(X,a)=0\), bilinearity
    • Variance of sums \(\text{Var}(aX\pm bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)\pm2ab\,\text{Cov}(X,Y)\)
    • Independence \(\Rightarrow\) zero covariance (converse fails)
    • Correlation as covariance of standardized variables
    • Mean-independence as an intermediate concept

Quiz

Before class, complete the review questions for class 3. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 2.

Flashcards

Download the Class 3 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number three. Over the vacation make sure to study for your collection on probability! This will take place in HT Week 0.

Date, Time, and Location

Thursday, 3:30-6pm, HT Week 2, Location TBC

Videos

  • S0: Statistics Intro
  • S1: Populations, Samples & Random Variables
    • Populations (enumeration vs. model); samples and representativeness
    • Sampling frames; simple random sampling vs. quota sampling; non-response/selection bias
    • Random variables as functions of uncertain events; \(X_1\) inherits the population distribution
    • iid; sampling with vs. without replacement; large-population approximation
    • Moments of iid samples equal population moments
  • S2: Parameters, Statistics & Estimation
    • Parameters vs. statistics; sampling variation
    • Estimand / estimator / estimate; notation (\(\hat\theta\), \(\bar X\), \(s^2\), \(\hat p\))
    • The analogue approach; standard estimators (mean, variance, covariance, correlation)
    • Sampling distribution of an estimator
    • Unbiasedness; bias of the sample variance and Bessel’s correction
    • Efficiency (\(\sigma^2/N\)); consistency; consistent-but-biased estimators
  • S3: The Law of Large Numbers & Central Limit Theorem
    • The single-sample inference problem; sampling distribution of the mean
    • LLN (Bernoulli 1713): \(\bar X_N \to \mu\) in probability
    • CLT (Laplace 1810): standardized mean \(\to N(0,1)\)
    • Implied approximation \(\bar X_N \sim N(\mu,\sigma^2/N)\)
    • Standardization and the location-scale transformation
    • Simulation illustration; LLN/CLT as foundations of inference
  • S4: Intro to Statistical Inference
    • Inference: recovering population parameters from sample data
    • Estimates vs. parameters; the sampling distribution
    • The two tools: confidence intervals and hypothesis tests
    • Parameter/estimator table (general vs. binary)
    • CLT-based normal approximation; plugging in \(s^2\) for \(\sigma^2\); nuisance parameters
  • S5: Confidence Intervals Overview
    • Deriving a CI for the mean from an iid sample
    • Standardizing and invoking the CLT; the 95% probability statement
    • General form: statistic \(\pm\) (number of std. errors) \(\times\) std. error; critical values
    • CI width vs. confidence level, std. error, variance, sample size
    • The CI as a statistic; correct vs. incorrect (frequentist) interpretation
  • S5a: Confidence Intervals for Means
    • CI for a single mean: \(\bar X \pm 1.96\sqrt{s^2/N}\)
    • Standard error of the mean; effect of \(N\) and variance on width
    • CI for the difference of two independent means
    • Paired (dependent) samples: \(\text{Var}(\bar X-\bar Y)=\text{Var}(\bar D)\)
    • Worked examples (anchoring experiment; paired exams)
  • S5b: Confidence Intervals for Proportions
    • CI for a single proportion: \(\hat p \pm 1.96\sqrt{\hat p(1-\hat p)/N}\)
    • CI for the difference of two independent proportions
    • Paired (dependent) proportions as a single sample of differences
    • Why paired binary differences are “trinary” \(\{-1,0,+1\}\)
    • Worked examples (Trump approval; partisan; smoking cessation)

Quiz

Before class, complete the review questions for class 4. These will be the basis of the quiz at the start of class. (Since you have a collection on probability in HT Week 0, there will be no “extra challenge” questions based on Class 3 on this quiz.)

Flashcards

Download the Class 4 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number four.

Date, Time, and Location

Thursday, 3:30-6pm, HT Week 4, Location TBC

Videos

  • S6: Tests of Statistical Hypotheses - Overview
    • Null vs. alternative hypotheses (about parameters, not statistics)
    • Type 1 and Type 2 errors; criminal-trial analogy
    • The five-step testing procedure
    • \(Z\) statistic \(=\) (statistic \(-\) hypothesized) / std. error; \(N(0,1)\) null
    • Critical values, \(\alpha\), confidence level; \(|Z|>c\) rule
    • \(\alpha\) as long-run Type 1 error rate; the replication crisis
  • S6a: Tests for Means
    • General test statistic form; the 5-step procedure for means
    • One-sample test for a single mean
    • Two-sample test for a difference in independent means; SE \(=\sqrt{s_x^2/N_x+s_y^2/N_y}\)
    • Paired test reduced to a one-sample test on differences
    • Worked examples (single mean; anchoring; paired exams)
  • S6b: Tests for Proportions
    • Two complications: variance derivable from the proportion; SE under the null
    • Single-proportion test using \(p_0\) for the exact SE
    • Two independent proportions: pooled estimator \(\hat\pi\) and its SE
    • Paired-proportions test on the difference variable
    • Worked examples (Trump approval; partisan; smoking cessation)
  • S7: Confidence Intervals and Hypothesis Tests Revisited
    • Duality between confidence intervals and hypothesis tests
    • The CI as the set of null values not rejected by the test
    • Two-sided z-test; finding the boundary null values
    • Deriving the 95% CI from the non-rejection region

Quiz

Before class, complete the review questions for class 5. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 4.

Flashcards

Download the Class 5 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number five.

Date, Time, and Location

Thursday, 3:30-6pm, HT Week 6, Location TBC

Videos

  • T0: Time Series Introduction
  • T1: Time Series Data
    • Time series: chronologically ordered, serially dependent observations; modeling and forecasting
    • Trends and log transformations; breaks; seasonality; cycles; noise
    • Correlation, lags (\(X_{t-h}\)) and leads (\(X_{t+h}\))
    • Autocorrelation \(\rho(h)\); bounds and \(\rho(0)=1\); less reliable at large lags
    • The correlogram
  • T2: Basic Operations
    • Notation \(X_t\); lags, leads, initial condition \(X_0\)
    • First difference \(\Delta X_t=X_t-X_{t-1}\); seasonal (fourth) difference
    • Per-period growth rate \(g\)
    • Compound average growth rate over \(h\) periods (not the arithmetic mean of growth rates)
  • T3: Deterministic & Stochastic Time Series
    • Deterministic vs. stochastic components
    • Linear trend \(X_t=\beta_0+\beta_1 t\); aperiodic deterministic signal (irrational frequency ratio)
    • Stochastic series; iid Gaussian noise as “as-if random”
    • Structural break (regime shift); stochastic trend (time-varying mean); changing volatility
  • T4: Stationarity
    • Stationarity as the time-series analogue of iid
    • Strong vs. weak stationarity (constant mean/variance; autocovariance depends only on lag \(h\))
    • Characteristics of a stationary series
    • Inducing stationarity: differencing, detrending, variance-stabilizing transforms
    • Detecting non-stationarity (formal tests vs. “eye-ball econometrics”)
  • T5: Some Time Series Models
    • White noise (iid, zero mean, serially uncorrelated); Gaussian white noise as “shocks”
    • Random walk \(X_t=X_{t-1}+e_t\); non-stationarity (\(\text{Var}\) grows linearly in \(t\)); high persistence
    • Random walk with drift
    • AR(\(p\)) and AR(1) \(X_t=\theta_0+\theta_1 X_{t-1}+e_t\), nesting white noise / random walk / RW-with-drift; role of \(\theta_1\)’s sign
  • T6: Mean Reversion
    • Stationarity of AR(1): constant mean \(\theta_0/(1-\theta_1)\) and variance \(\sigma^2/(1-\theta_1^2)\)
    • Stationarity condition \(|\theta_1|<1\)
    • Autocorrelation \(\theta_1^h\) (exponential decay; sign alternation when \(\theta_1<0\))
    • \(\theta_1\) equals the one-lag autocorrelation
    • Galton’s regression to the mean as a stationary AR(1)
    • \(\mathbb{E}[X_t|X_{t-1}]\) as a weighted average; mean-reversion intuition and applications
  • T7: Spurious Correlation
    • “Nonsense” correlations vs. spurious correlation (formal sense)
    • Famous nonsense correlations (tylervigen.com); Hendry (1980), Yule (1926)
    • Spurious correlation between two independent random walks
    • Simulation: independent \(N(0,1)\) random walks can show \(\rho=0.71\); correlation roughly uniform on \([-1,1]\)
    • “Correlation does not imply causation,” acute for non-stationary series; differencing/log remedies

Quiz

Before class, complete the review questions for class 6. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 5.

Flashcards

Download the Class 6 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number six.

Date, Time, and Location

Thursday, 3:30-6pm, HT Week 8, Location TBC

Videos

For this class there are three sets of videos: (1) my introduction to causal inference for a general audience, (2) the lecture videos:

  • C0: Introduction
    • Causal inference vs. statistical inference (distinct)
    • “Population first” approach
    • Roadmap: potential outcomes, selection bias, RCTs, internal/external validity, natural/quasi experiments, LATE
  • C1: Potential Outcomes
    • Neyman–Rubin potential-outcomes framework; outcome \(Y\), binary treatment \(D\)
    • Potential outcomes \(Y(0)\), \(Y(1)\); individual effect \(Y(1)-Y(0)\)
    • ATE \(=\mathbb{E}[Y(1)]-\mathbb{E}[Y(0)]\); ATT (conditioned on \(D=1\))
    • Switching equation \(Y=Y(0)(1-D)+Y(1)D\)
    • Fundamental problem of causal inference; the counterfactual
  • C2: Selection Bias
    • Observed comparison \(\mathbb{E}[Y(1)|D=1]\) vs. \(\mathbb{E}[Y(0)|D=0]\)
    • Fundamental equation: observed difference \(=\) ATT \(+\) selection bias
    • Selection bias defined; arises when treatment depends on potential outcomes
    • Worked examples (Oxford-vs-University-of-X earnings; Youth Training Scheme); positive vs. negative bias
    • Selection bias distinct from sample bias; not directly observable
  • C3: Randomized Controlled Trials
    • RCTs to avoid selection bias; \(D\perp\!\!\!\perp\{Y(0),Y(1)\}\)
    • Randomization balances characteristics on average
    • Independence from potential vs. realized outcomes
    • Mean-independence; selection-bias term driven to zero
    • Under randomization ATT \(=\) ATE \(=\) observed difference in means
  • C4: Internal and External Validity
    • Internal validity (credible for the sample) vs. external validity (extrapolates to the population)
    • Threats to internal validity: contamination, non-compliance
    • Hawthorne effect; placebo effect
    • External validity defined; internal does not imply external
    • Sample/population mismatch; enumerative vs. eliminative induction; spillovers; short durations/surrogate outcomes
  • C5: Natural and Quasi-experiments
    • Alternatives when RCTs are infeasible/unethical
    • Randomization as the identifying mechanism (independence of assignment from potential outcomes)
    • Natural experiments (nature-induced variation)
    • Quasi-experiments (institutions, laws, borders); Angrist & Lavy (1999) Maimonides’ rule
  • C6: Conditional Independence
    • Conditional Independence Assumption: \(D\perp\!\!\!\perp\{Y(0),Y(1)\}\mid X\)
    • Selection on observables (Angrist 1998 military-service example)
    • Conditional decomposition; CIA eliminates the conditional selection-bias term
    • Recovering ATE/ATT by averaging cell-specific effects (Law of Iterated Expectations)
    • CIA is untestable/contestable; common support and the curse of dimensionality
    • Simpson’s paradox; “bad controls” (conditioning on an outcome) and the gender-pay-gap decomposition

and (3) some additional videos of mine that complement the lecture videos:

Quiz

Before class, complete the review questions for class 7. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 6.

Flashcards

Download the Class 7 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number seven.